Sift flour, baking powder, and baking soda into medium bowl. Beat brown sugar and butter in large bowl until blended. Add eggs 1 at a time, beating well after each addition. Add flour mixture in 3 additions, beating until smooth. Mix in pureed date mixture (batter will be stiff).

Spread batter in prepared pan. Bake until cake is firm to touch and tester inserted into center of cake comes out clean, about 35 minutes. Cool in pan on rack.

Turn cake out onto work surface. Remove parchment. Trim 1/2 inch from short sides of cake and 1/4 inch from long sides. Using serrated knife, cut cake horizontally into 3 equal layers. Return 1 layer to same pan. Pour 1 cup sauce over. Top with second layer. Pour 1 cup sauce over. Top with third layer. (Can be made 6 hours ahead. Cover; let stand at room temperature.)